Problem 38 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}+y^{2}=0 $$

Step-by-Step Solution

Answer

The graph is a point at the origin and it is a function.

Step 1: Understand the Equation Form

The given equation \( x^2 + y^2 = 0 \) represents a geometric figure in the coordinate plane. This equation resembles the standard form of a circle equation which is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Analyze the Circle's Center and Radius

In the equation \( x^2 + y^2 = 0 \), it can be reinterpreted as \( (x - 0)^2 + (y - 0)^2 = 0^2 \). Thus, the center of the circle is at the origin \((0, 0)\) and its radius is zero.

Step 3: Interpret the Radius and Graph

A circle with a radius of zero collapses into a single point. Therefore, the graph of the equation is just a single point, specifically the origin \((0, 0)\).

Step 4: Determine if it is a Function

A relation is a function if each input value (\(x\)-value) maps to exactly one output value (\(y\)-value). Here, there is only one point \((0, 0)\), so for the input \(x = 0\), there is a unique output \(y = 0\). Therefore, this graph is a function.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph coordinates or geometric equations. It is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin, which has the coordinates (0, 0).

Here are a few key points about the coordinate plane:

• Each point on the plane is identified by an ordered pair of numbers, known as coordinates, such as (x, y).
• The x-coordinate tells us how far the point is from the y-axis, and the y-coordinate tells us how far the point is from the x-axis.
• This system allows for the mapping and visualization of equations and geometric figures, like circles.

Understanding the coordinate plane is crucial when dealing with geometric figures because it provides the foundational framework for graphing and analyzing shapes.

The Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on its circumference. It is one of the key components in the standard form equation of a circle, which is expressed as a^2 + b^2 = r^2 🐇 In this equation:

• The term r represents the radius of the circle.
• The terms (a - h) and (b - k) represent the horizontal and vertical distances between the center (h, k) and a point on the circle.

In simpler terms, if you imagine a circle, the radius is like a spoke on a wheel, perfectly straight, and it reaches from the center of the circle out to the edge. In the exercise, the circle's equation is x^2 + y^2 = 0. We see that the radius is zero, indicating a unique situation where the circle is essentially a single point at the origin.

Exploring Geometric Figures

Geometric figures refer to shapes or images that are defined or contained within the coordinate plane. They encompass a variety of shapes including points, lines, curves, and in this case, circles.

Important aspects of geometric figures:

• Circles are one of the most common figures and they can be represented mathematically through an equation.
• Each geometric figure has unique properties: for a circle, these include the center and the radius.
• Understanding its characteristics helps in identifying its graph and analyzing its features.

In the scenario examined, despite having the formula of a circle, the specific inputs in the equation x^2 + y^2 = 0 indicate that the figure is actually a point. This exemplifies how geometric figures can sometimes defy our usual expectations, showcasing the flexibility and variety within coordinate geometry.